A relative equilibrium in the \(n\) -body problem is where the \(n\)  point
masses keep a fixed relative position during their motions. Only a
special set of configurations of the \(n\)  bodies can maintain such
relative equilibria, these are called central
configurations. Central configurations are well-studied objects
(cf. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}), dating back to Euler and
Lagrange, a simple question whether the number of central
configurations is finite, for any given set of \(n\)  positive masses,
remain a major open problem in mathematics (cf. Smale
[6]}, Albouy and Kaloshin[9]},
Hampton & Moeckel [10]} ).
